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In particular, the Kazhdan--Lusztig polynomials of the graphic matroids of $T_n$ and $K_{2,n}$ are real-rooted. We also prove that, for $n\\ge2$, the common polynomial $Z_{T_n}(x)=Z_{K_{2,n}}(x)$ has $n+1$ distinct negative zeros. 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Zhang","submitted_at":"2026-08-03T14:28:37Z","abstract_excerpt":"Let $T_n=K_{1,1,n}$, and let $P_n(x)$ denote the Kazhdan--Lusztig polynomial of its graphic matroid. We prove that, whenever $n\\ge2$ and $0\\le\\lambda\\le n/2$, the polynomial $P_n(x)+\\lambda x$ has exactly $\\lfloor n/2\\rfloor$ zeros, all of which are negative and simple. In particular, the Kazhdan--Lusztig polynomials of the graphic matroids of $T_n$ and $K_{2,n}$ are real-rooted. We also prove that, for $n\\ge2$, the common polynomial $Z_{T_n}(x)=Z_{K_{2,n}}(x)$ has $n+1$ distinct negative zeros. 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